# MATH3962/4062 Rings, Fields and Galois Theory (Advanced)

## General Information

• Taught in Semester 1.
• Credit point value: 6.
• Classes per week: Three lectures and one tutorial.
• Lecturer(s): James Parkinson .

Please refer to the Senior Mathematics and Statistics Handbook for all questions relating to Senior Mathematics and Statistics. In particular, see the handbook entry for MATH3962 for further information relating to MATH3962.

You may also view the description of MATH3962 and the description of MATH4062 in the University's course search database.

For enrolled students or other authorized people only, here is a link to the Canvas page for MATH4062.

Students have the right to appeal any academic decision made by the School or Faculty: see sydney.edu.au/students/academic-appeals.html.

## Class and consultation times

• Lectures will be held on Mondays, Wednesdays and Thursdays at 14:00.
• My consultation time is Wednesday 1-2pm in Carslaw 614.

## Unit outline

This unit of study investigates the modern mathematical theory that was originally developed for the purpose of studying polynomial equations. In a nutshell, the philosophy is that it should be possible to completely factorise any polynomial into a product of linear factors by working over a "large enough" field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory.

The basic theoretical tools needed for this program include the fundamental concepts of groups, rings, and fields. The course begins with the definitions and examples of these concepts, as well as the associated structures such as subgroups, subrings, homomorphisms, ideals and quotients. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which links the process of solving polynomials with the process of taking field extensions, and then links this process to properties of the Galois group of the polynomial. Of course there is a lot to learn before understanding how this all works, but the point is that the problem of solving polynomial equations is converted into a much more accessible problem in finite group theory.

Along the way we will see some beautiful gems of mathematics, including Fermat's Theorem on primes expressible as a sum of two squares, solutions to the ancient greek problems of trisecting the angle, squaring the circle, and doubling the cube, and the crown of the course: Galois' proof that there is no analogue of the "quadratic formula" for the general quintic equation.

Here is a week-by-week plan of the topics that we will cover. However things might change, and the lectures are the definitive guide for the content of this course.

Week 1
Introduction and overview, definitions of groups and rings, examples
Week 2
Subgroups, cosets, Lagrange's Theorem, normal subgroups, quotient groups, the symmetric group
Week 3
Subrings, polynomial rings, homomorphisms, ideals, and the First Isomorphism Theorem for groups and rings
Week 4
The Correspondence Theorem, integral domains, field of fractions of an integral domain
Week 5
Principal ideal domains, Euclidean domains, greatest common divisors, prime and irreducible elements
Week 6
The Unique Factorisation Theorem, unique factorisation domains, case study: Gaussian integers.
Week 7
Unique factorisation in polynomial rings, irreducibility in polynomial rings
Week 8
Irreducibility in polynomial rings continued, ring and field extensions
Week 9
Minimal polynomials, degree of a field extension, constructible numbers
Week 10
Solution to constructibility problems, constructible polygons, splitting fields, separability
Week 11
Finite fields, Galois groups, statement of the Galois correspondence, the order of the Galois group
Week 12
Proof of the Galois correspondence, solving polynomial equations using radicals, insolubility of the general quintic
Week 13
Revision and tying off loose ends.
This is subject to change, depending on our progress and inspiration.

## Learning outcomes

The learning outcomes for this unit of study are as follows.

• be familiar with the basics of abstract ring and field theory;
• be familiar with the concepts of integral domains, principal ideal domains, Euclidean domains, and unique factorisation domains, and understand the relationships between these concepts;
• understand the concept of irreducibility in integral domains;
• be proficient at applying various irreducibility tests;
• be proficient at applying the Euclidean Algorithm in various contexts;
• have a solid working knowledge of the basic examples of rings and fields including the integers, Gaussian integers, polynomial rings, the rational numbers, and finite fields;
• be able to work with field extensions, including computing the degree of an extension and the minimal polynomial of a simple extension;
• understand the solutions to the three ancient greek geometric problems;
• know and be able to apply the basic concepts and definitions from Galois Theory;
• know the basic properties of the Galois group of a field extension;
• be able to compute Galois groups in simple examples;
• be able to construct proofs, including sophisticated proofs using a variety of concepts covered in the unit;
• be proficient in dealing in abstract concepts with an emphasis on the clear explanation of such concepts to others;
• be able to apply the theory and methods introduced in the unit to specific examples, both those encountered in lectures and tutorials, and to related examples.

## Assessment

• Assignment 1 was due Thursday 9th April. The questions are available here. The solutions are available here. You can also find some general feedback here. Please read the instructions on the assignment carefully. In particular, your assignment must be typed using LaTeX. Here is a template tex file that you may use.
• Assignment 2 was due Monday 25th May. The questions are available here. The solutions are available here. Please read the instructions on the assignment carefully. In particular, your assignment must be typed. Here is a template tex file that you may use if you like.

Your mark for MATH3962 will be calculated as follows (note that adjustments have been made as a result of moving online in response to covid19).

• Two assignments, worth 20% each. The assignments will give practice in investigating examples and constructing proofs, and feedback should help with your mathematical writing skills and exam preparation. The assignments are due (via LMS) by midnight on the following dates:
• Assignment 1 due on Thursday 9th April (Week 7)
• Assignment 2 due on Monday 25th May (Week 13)
• Final exam, worth 60%, during examination period. For further details, see the Exam Information section below.
• Tutorial participation is no longer an assessable component of thus unit. However we will move to weekly online zoom tutorials (Monday, 1pm). Mathematics is not a spectator sport. We will be working through the questions together. The tutorials are an integral part to the course, since the lectures are pretty dense and theory based. So it is very beneficial to attend our weekly meetings.

The tutorial sheets will be posted below. We won't get through all the questions in the tutorial. It is expected that you spend at least 3 or 4 hours of your own time each week finishing off as many of the questions as you can. This is key to success in this challenging course.

High Distinction (HD), 85-100
Complete or close to complete mastery of the material
Distinction (D), 75-84:
Excellence, but substantially less than complete mastery
Credit (CR), 65-74:
A creditable performance that goes beyond routine knowledge and understanding, but less than excellence
Pass (P), 50-64:
At least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

## Reference books

The content of the unit is defined by the lectures rather than by a set text. Even though there is no reference book for the course, students might find the following lecture notes from previous years helpful:

It is always a good idea to consult other sources for extra problems and alternative explanations. Most online mathematical encyclopedias contain material relevant to this unit. Be aware that conventions and notation may differ slightly from those in the lectures. The following books could be used to provide further practice if you like:

• Abstract Algebra, D. Dummit and R. Foote (this is an excellent reference, also for group theory)
• Galois theory, E. Artin
• A survey of modern algebra, Garrett Birkhoff and Saunders Mac Lane
• Modern algebra: an Introduction, John R. Durbin
• A first course in abstract algebra, John B. Fraleigh
• Abstract algebra, I. N. Herstein
• [ISGalois theory, I. N. Stewart

## Linear and Abstract Algebra Revision: Essential reading!

Below is a very useful summary of what you covered in MATH2922. It is absolutely essential that you have a good grasp of this material. In particular, you have to know all of the material on groups, fields, vector spaces, linear transformations, and matrix representations. This is the material up to and including Lecture 5-3 in the notes:

In particular, it is important that you have a good understanding of group theory. Here are some questions on group theory and here are the solutions.

## Lecture material

The following lecture notes are from the first three weeks, when face-to-face lectures were still running.

From Monday 16th March we begin our transition to online lecturing. During the "transition week" (week 4) the Echo360 lectures from 2019 were provided, along with a supplementary audio/pdf recording providing further context, examples, and motivation for the material.

From Monday 23rd March we have completely transitioned to online teaching. Each week will now consist of a series of "mini-lectures (between 4 and 8 each week). There is a single accompanying pdf, which contains the entire week's lectures in one file. In total, these mini-lectures allow us to provide more examples, motivation, and context than what could have been provided in the 3 hours of lectures - one of the few benefits of corona virus.

The video recordings are available on our CANVAS page (under "Lecture Recordings"). The pdfs are provided below (the links will become active after the material is uploaded as the semester progresses).

## Tutorial questions and solutions

Tutorials were to be held in Weeks 2-13 (so the first tutorial is in week 2). Of course covid19 changed this. From week 5 onwards, the tutorials are via zoom.

Tutorials questions and solutions can be downloaded below. All question sheets are active links, however the links to the solutions will only become active as the semester progresses:

Math3962/4062 Exam Information
• The MATH3962/4062 end of semester exam is scheduled for MONDAY 22 JUNE, 3pm - 6pm. You will be able to access the exam questions at 3pm via CANVAS, and you will have until 6pm to complete, scan/photograph, compile, upload, and submit your solutions via CANVAS.
• The front page of the exam is available here: Front page of exam, containing detailed exam instructions.
• Your uploaded exam solutions must be in the format of a SINGLE PDF file.
• Practice of exam logistics: On Tuesday 26th May we will have a practice of the exam download/upload process. Details will be sent via email/Ed.

Here is a selection of past exams (the more recent papers are perhaps more relevant in terms of content):

2019     2018     2017     2016     2014     2013     2011     2009

Solutions to the 2018 exam and selected questions from the 2017 exam are available here. See the Week 13 lectures for solutions to the 2019 exam. Further solutions will not be provided.

I will run two exam consultations, via zoom (these times are subject to change if there are significant clashes with your other exams):

Monday 15th June 11am-1pm (Zoom, see CANVAS for link)

Wednesday 17th June 11am-1pm (Zoom, see CANVAS for link)

## Timetable

Show timetable / Hide timetable.